The Transient Climate Response and the Equilibrium Climate Sensitivity

There seems to have quite a lot of dicussion recently on Watts Up With That (WUWT) about climate sensitivity, in particular the Transient Climate Response (TCR) and the Equilibrium Climate Sensitivity (ECS). Some of what has been claimed on WUWT has got me rather confused, so I thought I would write this post to clarify what I think and also to give others an opportunity to correct me if my understanding is wrong.

I’ll start with the TCR. It is defined here as

the change in the global surface temperature, averaged over a 20-year period, centred at the time of atmospheric carbon dioxide doubling, that is, at year 70 in a 1% per year carbon dioxide increase experiment with a global coupled climate model. It is a measure of the strength and rapidity of the surface temperature response to greenhouse gas forcing.

On a number of WUWT posts, it has been calculated using
TCR = F_2x ΔT/ΔF,
where F_2x is the change in forcing due to a doubling of CO₂ (typically calculated to be 3.7 W m^-2), and ΔT and ΔF are the measured changes in temperature and forcing over some time interval. Firstly, this kind of makes sense because it tells you, given a change in temperature ΔT due to a change in forcing ΔF, what the change in temperature will be once ΔF = F_2x. However, it does seem to be an approximation in that it doesn’t depend on the rate of increase of CO₂ while formally the TCR is defined as the temperature (averaged over 20 years) after the CO₂ has doubled at a rate of 1% per year for 70 years. Also, it seems like ΔF should be the change in forcing due to CO₂ only. If not, ΔF can be larger than that due to CO₂ alone and the TCR will equal ΔT before CO₂ has doubled (i.e., it will underestimate the TCR).

For example, in this post Willis Eschenbach attempts to calculate the climate sensitivity directly from model results using an equation of the form (I’m leaving out some terms, but this is the basic form)
T(n+1) = T(n) + λΔF(n+1),
where he refers to λ as the climate sensitivity. However, he seems to use the total change in forcing, ΔF, and so this would seem to be the climate sensitivity to an increase in total forcing of 3.7 W m^-2, rather than the sensitivity to a doubling of CO₂.

The ECS is defined according to this as

a measure of the climate system response to sustained radiative forcing. It is defined as the equilibrium global average surface warming following a doubling of CO2 concentration.

Again, this seems to be determined on a number of WUWT posts using
ECS = F_2x ΔT/(ΔF – ΔQ),
where the terms are the same as for the TCR and ΔQ is the rate at which the oceans are being heated. Again, this makes sense. If, after some time interval, the forcing has changed by ΔF with ΔQ going into heating the oceans and the difference (ΔF – ΔQ) going into heating the surface, then the measured change in temperature ΔT tells you how much this net forcing increases the surface temperature. Multiplying the by change in forcing caused by a doubling of CO₂ then tells you how much the temperature will eventually increase by if CO₂ will double.However, this again seems like an approximation and that to determine the ECS properly probably needs detailed climate modelling.

Also, it seems like ΔF and ΔQ should, again, only be the contributions due to CO₂. If not, then it would seem like the ECS is an estimate of the change in temperature if the total forcing changes by 3.7 W m^-2, rather than if the CO₂ concentration doubles. I would be interested to know if my understanding is correct, so if anyone knows more about this than me (of which there must be many) I’d be keen for them to clarify. It also seems, if my understanding is roughly correct, that the reason a number of the posts on WUWT are getting lower climate sensitivities than other work is because they’re using the wrong values for ΔF and ΔQ – although it would be good if someone could clarify whether this is the case or not.